Question

If \(\alpha, \beta\) are two distinct eigenvalues of a matrix \(A\), then the corresponding eigenvectors are

• A. Linearly dependent
• B. Linearly independent
• C. Orthogonal
• D. Orthonormal

Hint:

Always remember: \[ \text{Distinct eigenvalues} \Longrightarrow \text{Linearly independent eigenvectors}. \] Orthogonality is guaranteed only for special matrices such as symmetric or Hermitian matrices.

Answer 1

The Correct Option is B

Concept: One of the most important results in Linear Algebra states that eigenvectors corresponding to distinct eigenvalues of a matrix are always linearly independent. This theorem helps in diagonalization of matrices and plays a fundamental role in the study of linear transformations. It should be noted that orthogonality is guaranteed only for certain classes of matrices such as symmetric or Hermitian matrices. For a general matrix, eigenvectors corresponding to distinct eigenvalues need not be orthogonal.

Step 1: Assume two distinct eigenvalues.
Let \[ A\mathbf{x}=\alpha\mathbf{x} \] and \[ A\mathbf{y}=\beta\mathbf{y}, \] where \[ \alpha\neq\beta. \] Here \(\mathbf{x}\) and \(\mathbf{y}\) are eigenvectors corresponding to eigenvalues \(\alpha\) and \(\beta\), respectively.

Step 2: Assume they are linearly dependent.
Suppose \[ \mathbf{y}=k\mathbf{x} \] for some scalar \(k\neq0\). Applying \(A\) on both sides, \[ A\mathbf{y}=A(k\mathbf{x}) \] \[ =kA\mathbf{x} \] \[ =k\alpha\mathbf{x}. \] Since \(\mathbf{y}=k\mathbf{x}\), \[ A\mathbf{y}=\alpha\mathbf{y}. \]

Step 3: Compare with the eigenvalue equation.
But we are also given \[ A\mathbf{y}=\beta\mathbf{y}. \] Therefore, \[ \alpha\mathbf{y}=\beta\mathbf{y}. \] Since \(\mathbf{y}\neq0\), \[ \alpha=\beta. \] This contradicts the given condition that the eigenvalues are distinct.

Step 4: Draw the conclusion.
Hence our assumption was wrong. Therefore the corresponding eigenvectors cannot be linearly dependent. Thus they must be linearly independent. \[ \boxed{\text{Eigenvectors corresponding to distinct eigenvalues are linearly independent}} \] Hence option (B) is correct.